https://file.aiquickdraw.com/mathbot/file/1727389603803d09rc5ws.jpg

This problem involves finding the area under the normal distribution curve between two given values, which represent exam scores on a standardized test.

Given dataThis problem involves finding the area under the normal distribution curve between two given values, which represent exam scores on a standardized test.

Given 
- Mean (\(\mu\)) = 538
- Standard deviation (\(\sigma\)) = 50
- The two values on the graph are 504.5 and 588.

To find the area under the curve (the probability), we follow these steps:

1. **Convert the scores to z-scores**:  
   The z-score is calculated using the formula:
   \[
   z = \frac{X - \mu}{\sigma}
   \]
   where \(X\) is the score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

2. **Find the cumulative probabilities** for these z-scores from a standard normal distribution table or using a calculator.

3. **Subtract the cumulative probabilities** to find the area under the curve between the two z-scores.

Let me calculate that for you.The shaded area under the curve between the scores of 504.5 and 588 is approximately **0.590** (to three decimal places).

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What does a z-score represent in a normal distribution?
2. How does changing the standard deviation affect the shape of a normal distribution curve?
3. How can you use the z-score to compare scores from different distributions?
4. What is the total area under a standard normal curve?
5. How do you calculate probabilities for values not exactly at 1, 2, or 3 standard deviations from the mean?

**Tip:** The area under the normal distribution curve between two points represents the probability that a randomly chosen value from the distribution will fall within that range.

Scores on a standardized exam are normally distributed with a mean of 538 and a standard deviation of 50. The figure shows the distribution of scores on a standardized exam with the values 504.5 and 588. Calculate the shaded area under the curve.

This problem involves finding the probability between the scores of 504.5 and 588 on a standardized exam with a mean of 538 and a standard deviation of 50. The problem uses key concepts like z-scores and the standard normal distribution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation